[Music] [Music] [Music] you hi all my name is Ray Prakash and welcome to this second class of geometry okay so from here the real Jupiter will start right we'll do each and every concept of triangles and circles because mostly the questions in this example a cat ends at our visual triangles and surface right so we'll discuss each and every concept okay restart so most you will do triangles at first and then we will do circles okay so let's start with Travis see in Travis I'll first start with all the all the points containing geometric centers okay so all the points containing the geometric centers which which point should come under the geometric centers right very important okay please pay attention to matrix centers what is this now there is basically for geometry centers in triangles right first one is first one is centroid centroid okay notice centroid here centroid is basically intersection point of inter section point of all the medians okay so it is intersection point of all the videos right now what are medians what are medians so again see used it many many terms like this maybe you're not familiar with average of material like that right so media is right what it means now so media's contain a word mid minimis twining line joining me to point okay so you remember like this it's okay so medians contain world mid mid means 20 middle pointer edge what am i D is basically see you take a pic of a triangle like this triangle ABC so in triangle ABC now let us say VC midpoint is D okay so D is the midpoint of VC that means BD and DC will be equal okay if you join ad here this ad is called the median right this ad is called the video so here ad is the fine so similarly I can draw three minutes in triangle one is ad another is line joining midpoint of a and C so let's say B e okay is the midpoint of a and C that means a e and E C are equal okay so I can join ve sorry so I can join ve and then I can join and this let us say F is the midpoint of a B so I can join CF as well right so they will they will pass through a common point this is my construction right for sure nail passed through a common point this point is called centroid so all three will pass through a common point and this point is called one this point is called centroid centroid okay so with this image in mind read a triangle with all the line joining the midpoints okay and the intersection point is centroid right so what is the property is is intimate so properties of the centroid are centroid divides median centroid divides the median in the ratio 2 is to 1 okay 2 is to 1 where 2 is the part from vertex right - is the part from vertex so - this basically if it is a line and if this is a centroid okay this will divide this line in the ratio 2 is to 1 that means this this part is 2 and this part is what the ratio is 2 is 1 ok let name this lips name this point as its am so let's the M is centroid ok that means basically km/h to M be the ratio is what 2 is 2 okay 3 shows what 2 is 2 fine similarly for line ve so this vertex part is order to write this M in central idea this word X part is close this part is 2 and this part is 1 3 2 is 2 is 2 1 fine so that voltage a B and C are the vertex of the triangle a B and C are the voltage of the transistor ok centroid divides median in the ratio to this 2 1 now next point when you draw centroid rate so you can see here in this triangle right now there are six triangles one two three four five and six right so triangle is divided into six parts right so that means I can say if we draw all the medians so triangle is divided into six triangles of six triangles of equal even right so right now I can take it as a property of center at this point later on we need when you do the ratio method of triangles area division right then you can easily prove this right so why it is so ok so right now again take it as a property of straight divided when this is the centroid so it red comes by joining all these three medians so triangle is divided into six triangles of equal area okay and I mean we will do a ratio method of area division can easily prove this point right void is of six triangles of equal area fine so you can remember is three points in okay in centroid okay let's move to net point next a metal center so next two matrix center will be orthocenter okay so next joint isn't is also Center on center now what is also center and out all strip properties will do okay so orthocenter is basically intersection point of all the altitudes of the triangle okay BC it is it is what it is the intersection point of all the altitudes altitudes are also called heights of the triangle rate so all the heights of the travel fine so also Center is intersection point of all the heights of the turn rate let's draw it let's draw it see it is a triangle ABC okay and drawing a perpendicular right this is the height so let us say D is the height similarly I can draw a perpendicular on EC also so here we will be the it--let's thing ve is the I and again let us say CF is the ID if I draw a perpendicular on TV it says CF is the right right all the perpendicular this is sine of 90 degree right so now this is the intersection point right again this by construction all the three point lines has to be have to meet at a common point right this is called the orthocenter right they named this point as o and this point is called the orthocenter of this translation right so again you can remember like know ortho ortho is a root word right in English so or thumb is what also is the straight right so which then will be spread perpendicular tangent which trade right so ortho means is parade in verbal since reading is a root words right so it's been straight line okay so this is a straight line DD's and straight line B is a straight line right it doesn't matter like triangle can be like this also this is a triangle ABC so here ad is the height of a straight line D D is the right and what is the median is a midpoint okay so it's ABC midpoint is e so now a E is the median and EDD is the iDrive so they will not be same okay they'll still be same look in one his special case of isosceles triangle and one and in equilateral triangle right so well shut is at that point later on okay so that that way is here this is a also Center what is also standard intersection point of all the altitudes are height of the trend right now a very important property of all also Center lies here right a very important property of also Delta C the property is if there's a triangle ABC ok triangle ABC and let's say o is the orthocenter Oh is the orthocenter fine for is the off centigrade just make this image in mind okay oh is the oscillator then angle B or C plus its opposite angle at the vertex red so angle B or C plus X up plus it's a polar angle is angle T it's sum will be 180 degree right very important point and will be over C plus angle a is equal to 180 degree okay for does angle you see and they're made by Auto Center within two vertices right so obviously third vertex will be left that angle of also centered with the third vertex that is the opposite vertex that sum is always 180 degree right very important point okay now I can make such three again I can make another vertex here right so I can again meet like okay see is a triangle so again the triangle ABC and now let's say again always Auto Center okay so now let us say adjoining Stevo and Abel right so again this is the Auto Center it it is making this angle a GoSee with the other two vertices right so again what is the probability this angle of also centre plus its opposite angle at a vertex okay so again this sum is what 180 degree fine again that angle a Oh C+ and will be that sum is 180 degree right that means these two angles are supplementary okay so and then made by also centre with the two vertices and its opposite angle at the what third vertex right both this angles about supplement a supplemental is what some of fan and this is 180 D okay so very important point right frame at this point very wanted I can prove it also right it's quite a easy to prove also okay you want me to prove it I'll prove it right but see ingenuity where we cannot prove every point right there is so many theorems here so many theorems using so many a result side so it's not possible to prove every points otherwise ready to battery for only hundred hours right so I discuss all the points whatever point is relevant pause it for the examinations right if you want any proof of it you just simply google it or a certain NCR debauch right so kindly and I will personally tell you to avoid deep oops right don't ever always like okay this is its formula what is the proof right no cat is and cat and zeager AB PQ resignations right they have to use that formula huh that is like one concept right to this concept combined with other two three concepts we form a single question so cat is all about application of the concepts right so especially in geometry so here in Norway not needed to prove it right but okay it's a small group and it will help you also right so wherever it is required and obviously prove that results also right okay let me prove it quickly make a triangle ABC with limited angle ABC let me draw all the altitudes now ve is the altitude here CF is the order Judea and AD is the auditorium right PD is the auditorium fine now see this this is the Auto Center wicked oh okay see now look at triangle look at triangle VEC look at triangle BDC now I can write that let this angle be theta writing this angle s theta I am marking this angle is Phi okay this angle is Theta this angle is Phi and then e BC is theta and then f CB is Phi okay now what is that in look at triangle BC in triangle B EC I can write like okay this theta is equal to theta is equal to 90 minus angle C fine I can write because this is 90 this is this whole is angle C so obviously this is a theta is a third angle rotor angle before some to be 180 if this angle is 90 this is theta so this is angle C so what is Theta 90 base angle C similarly now look at triangle FCB look at triangle FC be inter angle FCB again so off 300 this angle is 90 degree fine this is Phi and this is whole and will be cool and will be right so angle B plus angle Phi plus 90 has to be 180 degree because it dragon sum of three angle just 180 degree okay so here I can again write what is Phi is equal to the Phi is equal to 90 minus and then V it is 90 minus angle V right now now look at triangle B or C look at triangle B or C I get a triangle b OC theta plus Phi is equal to 180 minus angle B you see I can write like this okay I can write like this that theta plus Phi is equal to 180 minus angle B or C okay why because some again some is 180 degree theta plus Phi plus angle B OC sum has to be 180 degree right now from these two equation you can add and replace the value here right we have the value theta plus Phi so you can add these two equations so if you add these two equations result will be theta plus Phi will be equal to 180 minus angle B minus angle C if I add these two equations fine we'll put just three SS five here so while replacing again theta plus Phi here this equation looks like now theta plus Phi is 180 minus angle B minus angle C right theta plus Phi I am replacing with this is equal to 180 minus and the V okay is equal to 180 minus and then we use now 180 180 gets cancelled that means angle B or C is equal to angle B plus angle C and I can write angle B or C is equal to 180 minus and lt right because in this bigger triangle ABC and will angle A plus angle B plus angle C will be equal to 180 degree so the value of angle B plus C so B plus C we divide 180 minus angle D 180 minus angle which it proves that and they'll be OC is equal to 180 minus angle a therefore I can write therefore I can write and we'll be oc+ angle a is equal to what 180 degree right so this is a proof for it right solid number eight in it and orthocenter angle plus its opposite vertex angle sum is always 180 right we'll do pushes right so we'll get clarity there also so remember this fine okay move to third third riveted Center okay now the third geometric Center is basically in the center third geometric Center is what in center now what is in center in center is basically intersection point of all the angle bisectors right so in st. is basically intersection point of all the angle bisectors now what are bisectors bisector means cutting in two equal halves right so obviously what angle bisectors cut cutting the angle into two equal half right so let's draw that let me draw angle ABC so a triangle ABC in this let us say now let us say T is the angle bisector that means this angle and this angle is same now again I can write B E is the angle bisector of angle B that means this angle and this angle is same and again it's is CF is the angle bisector of angle C that means this angle and this angle are C right CF is the angle bisector okay now this again by construction for sure they'll meet at a common point okay and this common point is basically called the in-center it is called the in-center okay this common point is called the in-center where they will intersect so it is in center okay now what is this property of instead of discuss right again there are a few important things this relates to instant death okay so what is the in-center property so in a center property is again I can write like angle and they'll be icy angle b IC is equal to ninety plus half of angle a right similarly angle a ib is equal to ninety plus half of angle C and angle a IC is equal to ninety plus half of and we'll be right there three very important points very important point right again prove it right because a small proportion and at concept will use for the focus I'll prove it so and then first we give me results so it's easy to remember also what is angle B I see so in angle B I see this I will be I see red is ninety plus half of the opposite angle at the vertex right so angle B I see a put endless word upon angle is angle a so angle B is equal to ninety plus half of angling again what is angle a I see in angle a AC again ninety plus half of the opposite angle at the vortex so ninety plus half of angry now okay let's prove it quickly improve it see look at triangle B I see I can write that in triangle be I see in triangle B I see I can write now angle Don since this is an angle bisector like a night this is this angle is C by 2 this angle is C by do it this angle is C by 2 CC by 2 similarly this is B by 2 this is p by 2 right so now you can see that in angle BAC again in triangle B I see some of these three angles will be equal to 180 degree okay so that is angle b IC is equal to 180 minus b by 2 plus angle B by 2 plus and the C by 2 I can write this simply because some has to be 180 degree right let this be equation 1 now replace the value of angle B by 2 plus angle C by 2 by B the triangle right so in bigger triangle a B second right so in bigger triangle ABC if angle A plus angle B plus angle C is equal to 180 degree since then and I need to get the value of B by 2 and C by 2 because already in terms of V I see only ok so how to replace this with so divide both sides by 2 so angle a by 2 plus angle B by 2 plus angle C by 2 is equal to 90 degree and then C by 2 is equal to 90 degree okay 90 degree dividing both sides by 2 what is angle what is angle B by 2 plus angle C by 2 is 90 minus a by 2 can replace here right replace is here you simply put 180 minus 90 minus a by 2 this is nothing but 90 plus and then a by 2 right so you can see you angle b IC is equal to 90 plus angle a what is the first result and the 90% in avoiding that is half of angle a right so you can prove this so good result and this is the property of in center right now one more very important tip on property of in center lysee phrase if you are a bit of that from in center we can form an incircle right and next point in user comes and Tessa from circumcenter is Rama we can form circumcircle so very important point this one from in center we can form a in circle right so let's again draw an angle EB C so how did was it inside the C triangle ABC let IBD in center okay so if I is the in center so keeping eye as this Center and keeping eye as the center and taking one perpendicular on any side and keeping here the compass and keeping here the pencil compass here at eye and pencil here I can draw a circle like this which will touch this circle at exactly three points at exactly three points right just not perfect but also the unit circle it touching is that is a see the tiny right this circle is called what this circle is called in circle this circle is called in circle and it's radius small R is called the inner radius in gym floor geometry will use a small R for in radius and capital R for circumradius circle it is able to discuss in next slide okay so in circle and what is small R it is in radius okay it small R is what it is in radius this is in it is in circle right so what is encircled it will touch this circle at exactly three points from inside right exactly three points this is the in circle and its 3ds is called in radius right so why drive it perpendicular here why ever have a bit perpendicular only you write three days because BC is the tangent to the circle okay because BC is a tiny to the circle and in it's a property of circle dad radius and tangent of circle always makes an angle of 90 degree okay that means if this is this circle okay and this is the tangent so this is not a tangent was not touching it and this is the tangent so touching the circle at just one point okay so radius and tangent makes an angle of what 90 degree okay so again it's a very important point 90 degrees our radial tangent 90 degree right that's why BC is a tangent here and this is the inner radius so there's our angle 90 degrees angle 90 degree okay so again a very important point within Center okay now next circumcenter fourth and last rivet resented his circumcenter now again a root word right in center in in center it contains in English word inside the circle is inside in circumcenter it contains a root word sir come ready sir come sir come basically means outside sir come is outside right that means a circle will form will deform outside the triangle right okay so do circumcenter the circumcenter is nothing but intersection point of intersection point of all the perpendicular side bisectors all the perpendicular side bicycle is right so that basically means that if I draw a triangle here triangle ABC triangle ABC and if this is a circumcenter let the circumcenter here is represented by okay let's say we set up into here so how do we do certain sentence I consider we'll get by this this will be as perpendicular bisector of BC right let's say this is the perpendicular bisector of easy okay this line is making 90-degree as well as bisecting VC okay making 90-degree as well as bisecting VC fine now again they leap up into the by the trove AC as well the perpendicular bisector of AC as well right so again this line is making 90 degree with AC as it is bisecting in two equal halves then is a perpendicular bisector of a B as well right so again this line will be like bisecting by setting this a be making 90 this towards an equal right there is a bit rough right so don't prove a diagram okay this is 90 degree sorry this is all circumcenter so intersection point of all the three perpendicular side by sector perpendicular side basically right this circumcenter what is the probability prop its property is that if this is circumcenter then keeping this point is called com+ and taking any one vertex as the radius this is a capital R right this capital R so again I should write here that this is point is the circumcenter okay and capital R is the circum radius circle radius okay so because circumcenter is I should write here that circumcenter is is equidistant is equidistant from all three vertices right the circumcenter is equidistant from all three vertices similarly I should say in in Center introduced either in center right so in center is equidistant from all the three sides should right here also right in center is equidistant equidistant from all three sides okay why because if you draw this circle here this is the inside this is the in circle right this is Center this perpendicular is also the radius this perpendicular is also the radius and this perpendicular to the radius right so all these three are equal okay that's why it is it credited all the three sides okay so so keeping the compas at this point at OU and taking any one vertices radius since all the three vertices are at equal equidistant from the circle Dilton all right area in the first point I can draw a circle which will touch this circle triangle from exactly outside and at exactly three points right exactly three points let me draw another one okay so exactly three points like this okay this is what this is the circum center circle circum circle means circum is outside so outside this circle outside the circle right now one important property of circumcenter circum a center is circumcenter if this o is a center here right if is a center and if I join this c o divine this see already social circle radius and if I joined this a o G of this also circumradius side so if this angle a is Theta understand right if this angle E is Theta if this angle is Theta then this angle b OC will be equal to 2 theta right this angle a is Theta so this angle b OC is equal to theta y y you can remember is talk about your circuit right if we remind you the property of circle was the property of circle was a chord a chord which forms a certain angle at the centre is double the angles formed by that chord at the circumference of the circle right that means let me write in your diagram diagram I can make like see in any circle this is a circle here right these are called any line digit inside a circle is called a god right any line inside a circle is called a coordinates the center so the first property is that if the angle formed add the voltage is Theta okay so angle formed by the same called at his Center will be 2 theta double of it right this is what is depicted here right because this is a center of the circle this BC is a chord here BC is the chord here okay so BC is forming theta decide to to data center so Devon theta circumference right so this two and is a double of it - Devon okay okay so one more providing and civilian disconnect rates right now if V is equal so a B in the same segment right this is one upper segment is the lower segment right in the same segment all the angles found by called a will be same that means this angle is Theta this angle is also theta okay this angle is also right all the angles the same right and its center it will form at Center it will form two theta right so very important points related to the metal center right these are the fortuity centers now we'll discuss its application some theorems in the next video okay thank you for watching q 82 [Music]